Fractal images are fun and you can download several free fractal generation programs on the Internet.
Fractals are a new discovery. Without computers nobody would have discovered them and without computers nobody would be looking at them.
Fractal math resulted in the development of Iterated Function Systems (IFS) that map a region of two dimensional space onto itself using a very small set of functions. The information needed to generate this fern, for example, is in a "seed" file with only 7 numbers in four sets (28 numbers total). Each set has 4 matrix numbers, two vector numbers and a single probability number.
A whole forest has been "mapped" onto a file of just 300,000 bytes. In any case, fractals are a very useful metaphor for understanding life.
Individual concepts - either thoughts or creatures - are like part of a fractal curve. One part of the design has an individual shape, with similar ones nearby. Each part is clearly different from its surroundings: different from the larger design it is part of, different from the smaller designs of its fine detail. But it does not exist by itself.
The existence and resolution of the fractal design depends on how many times the computer solves the fractal equation. If the equation is solved only a few thousand times, the design may not be visible at all.
Fractals are the surprising patterns that develop by graphing formulas of complex numbers on the two coordinates of your computer screen. The x axis (horizontal) represents real or ordinary numbers.
The y axis (vertical) represents imaginary numbers (real numbers multiplied by the square root of minus 1).
In a metaphoric sense, this resembles the creation of awareness and its subsequent remaping of probabilities. Biological systems are exposed to real numbers. They "map" these real numbers - expecting that future real numbers will be equal to those already experienced. The mapped numbers held in memory are imaginary, in the sense that they are expectations or representations of real numbers and exist on an imaginary plane. When imaginary numbers and real numbers turn out to be the same (or close enough not to trigger the sensory system), the result is nothing. No reaction, no dot on the display screen. In the Fractal shown to the left, no reaction is represented by black.
In the Mandelbrot fractal images no dot is represented by blue. When the expected number and the real one are not the same, the difference generates a point of awareness. The degree of difference between expected and real determines the degree of response. Because the sensors are all in different positions relative to the incoming signals, their combined flow of binary responses impose a second level of iteration in the next highest portion of the parallel computer - for example passing the information to processor clusters in the brain.
The internal computations are all very simple equations done on binary (yes-no) numbers and repeated millions upon millions of times per second. The operation of the retina of the eye is a reasonable example of what might be called parallel processing of fractal images.
A sea shell's decision to turn left or right is also based on hundreds or thousands of iterations of incoming sensory information matching real numbers against imaginary remembering. Threading into the equation is the internal state of the sea shell - hungry, frightened, sexually stimulated, searching for darkness or the light. The fractal equation of awareness, the constant remaping of real and imaginary based on surprise, has been solved by the experience of awareness learning new ways to minimize the error of expectations over several billion years.
Any one part of the structure of life on our planet - you or me, for instance - is clearly different from its surroundings and from the larger and smaller designs it is part of. But if we look closely enough at ourselves we find we are comprised of little iterations of becoming and if we look around ourselves we see we belong to larger iterations of becoming. And just like the fractal designs, no matter how closely you look, each pattern is similar, yet different from any other one.